orderly number

An orderly number is an integer $n$ such that there exists at least one other integer $k$ such that each divisor of $n$ from 1 to $d_{\tau (n)}$ (with $\tau (n)$ being the number of divisors function) satisfies the congruences $d_i\equiv 1modk$, $d_i\equiv 2modk$ through $d_i\equiv \tau (n)modk$. For example, 20 is an orderly number, with $k=7$, since it has six divisors, 1, 2, 4, 5, 10, 20, and we can verify that

$$1\equiv 1mod7$$

$$2\equiv 2mod7$$

$$10\equiv 3mod7$$

$$4\equiv 4mod7$$

$$5\equiv 5mod7$$

$$20\equiv 6mod7$$

The orderly numbers less than 100 are 1, 2, 5, 7, 8, 9, 11, 12, 13, 17, 19, 20, 23, 27, 29, 31, 37, 38, 41, 43, 47, 52, 53, 57, 58, 59, 61, 67, 68, 71, 72, 73, 76, 79, 83, 87, 89, 97, listed in A167408 of Neil Sloane’s OEIS. With the exception of 3, all prime numbers are orderly.